MHB Time-Discrete Odometry: Solving Complicated Equations By Hand

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The discussion revolves around solving equations related to time-discrete odometry in a cognitive robotics class. The user initially struggles with complex equations and seeks clarification on whether they are applying the correct formulas. A participant points out a sign error in the user's calculations, leading to a correction in the angle calculation. The conversation also touches on the use of polar coordinates to simplify the problem. Ultimately, the user resolves their confusion after identifying their mistakes with the signs in the equations.
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Hello, I have another question regarding my cognitive robotics class. Here is my current task:
View attachment 6205
And this is the slide from the lecture about this topic:
View attachment 6206
Now just plugging in everything I know into the formulas, I get this:
$$\boxed{3 = \sqrt{x'^{2} + y'^{2}}\\
-20 = atan2(y',x')\\
-30 = \theta' - 20}\\
\boxed{y' = \sqrt{9-x'^{2}}\\
-20 = atan2(\sqrt{9-x'^{2}},x')\\
\theta' = -10}\\
\boxed{y \approx 1.026\\
x = 3 \cos(20) \approx 2.819\\
\theta' = -10}$$

But then, when I try to perform the second motion, I get the following:
$$\boxed{10 \approx \sqrt{(x' - 2.189)^{2} + (y' - 1.026)^{2}}\\
20 \approx atan2(y' - 1.026, x' - 2.819) + 10\\
10 = \theta' + 10 - 20}$$
which is just way too complicated for me to solve (and we are supposed to solve this by hand).

Am I doing this right at all? Or is there some other formula that I should use? I haven't been able to find anything that helps me using Google :(
 

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Re: time-discrete odometry

RandomUserName said:
Now just plugging in everything I know into the formulas, I get this:
$$\boxed{3 = \sqrt{x'^{2} + y'^{2}}\\
-20 = atan2(y',x')\\
-30 = \theta' - 20}$$

Hi again RandomUserName! (Smile)

Let's take a look at this first step.
First off, shouldn't it be $-30 = \theta' - -20 = \theta' + 20$? (Wondering)

Okay, now for an intermezzo about polar coordinates.
Suppose we have the following right triangle:
\begin{tikzpicture}[ultra thick, font=\Large, blue]
\draw[thin] (4,0) rectangle +(-0.3,+0.3);
\draw (0,0) -- node[below] {$x$} (4,0) -- node
{$y$} (4,3) -- node[above left] {$r$} cycle;
\node[above right, xshift=4mm] (0,0) {$\alpha$};
\end{tikzpicture}
Then we have:
$$r=\sqrt{x^2+y^2}\\ \alpha = \operatorname{atan2}(y,x) \\ x=r\cos\alpha \\ y=r\sin\alpha$$

Can we find $x$ and $y$ from $r$ and $\alpha$ in the problem at hand? (Wondering)​
 
Re: time-discrete odometry

I like Serena said:
Hi again RandomUserName! (Smile)

Let's take a look at this first step.
First off, shouldn't it be $-30 = \theta' - -20 = \theta' + 20$? (Wondering)
Yes, you are correct, I messed up the sign there => $$\theta' = -50°$$
I like Serena said:
Okay, now for an intermezzo about polar coordinates.
Suppose we have the following right triangle:
\begin{tikzpicture}[ultra thick, font=\Large, blue]
\draw[thin] (4,0) rectangle +(-0.3,+0.3);
\draw (0,0) -- node[below] {$x$} (4,0) -- node
{$y$} (4,3) -- node[above left] {$r$} cycle;
\node[above right, xshift=4mm] (0,0) {$\alpha$};
\end{tikzpicture}
Then we have:
$$r=\sqrt{x^2+y^2}\\ \alpha = \operatorname{atan2}(y,x) \\ x=r\cos\alpha \\ y=r\sin\alpha$$

Can we find $x$ and $y$ from $r$ and $\alpha$ in the problem at hand? (Wondering)​

Omg, the more complicated the task gets, the more simple stuff I seem to not think of :D

edit:
Ok, I figured out my mistake, was another sign error... But now I'm good. Thank you again!​
 
Last edited:
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